This dissertation studies statistical decision making in various guises. I start by providing a general decision theoretic model of statistical behavior, and then analyze two particular instances which fit in that framework. Chapter 1 studies statistical decision theory (SDT), a class of models pioneered by Abraham Wald to analyze how agents use data when making decisions under uncertainty. Despite its prominence in information economics and econometrics, SDT has not been given formal choice-theoretic or behavioral foundations. This chapter axiomatizes preferences over decision rules and experiments for a broad class of SDT models. The axioms show how certain seemingly-natural decision rules are incompatible with this broad class of SDT models. Using those representation result, I then develop a methodology to translate axioms from classical decision-theory, a la Anscombe and Aumann (1963), to the SDT framework. The usefulness of this toolkit is then illustrated by translating various classical axioms, which serve to refine my baseline framework into more specific statistical decision theoretic models, some of which are novel to SDT. I also discuss foundations for SDT under other kinds of choice data. Chapter 2 studies statistical identifiability of finite mixture models. If a model is not identifiable, multiple combinations of its parameters can lead to the same observed distribution of the data, which greatly complicates, if not invalidates, causal inference based on the model. High-dimensional latent parameter models, which include finite mixtures, are widely used in economics, but are only guaranteed to be identifiable under specific conditions. Since these conditions are usually stated in terms of the hidden parameters of the model, they are seldom testable using noisy data. This chapter provides a condition which, when imposed on the directly observable mixture distribution, guarantees that a finite mixture model is non-parametrically identifiable. Since the condition relates to an observable quantity, it can be used to devise a statistical test of identification for the model. Thus I propose a Bayesian test of whether the model is close to being identified, which the econometrician may apply before estimating the parameters of the model. I also show that, when the model is identifiable, approximate non-negative matrix factorization provides a consistent, likelihood-free estimator of mixture weights. Chapter 3 studies the robustness of pricing strategies when a firm is uncertain about the distribution of consumers' willingness-to-pay. When the firm has access to data to estimate this distribution, a simple strategy is to implement the mechanism that is optimal for the estimated distribution. We find that such an empirically optimal mechanism boasts strong profit and regret guarantees. Moreover, we provide a toolkit to evaluate the robustness properties of different mechanisms, showing how to consistently estimate and conduct valid inference on the profit generated by any one mechanism, which enables one to evaluate and compare their probabilistic revenue guarantees.

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This dissertation explores the intersection between the fields of colloid science and statistical inference where the stochastic trajectories of colloidal particles are captured by video microscopy, reconstructed using particle tracking algorithms, and analyzed using physics-based models and probabilistic programming techniques. Although these two fields may initially seem disparate, the dynamics of micro- and nano-sized particles dispersed in liquids at room temperature are inherently stochastic due to Brownian motion. Further, both the particles under observation and their environment are heterogeneous, leading to variability between particles as well. We use Bayesian data analysis to infer the uncertain parameters of physics-based models that describe the observed trajectories, explicitly modeling the hierarchical structure of the noise under a set of varying experimental conditions. We set the stage in Chapter 1 by introducing Robert Brown's curious observation of incessantly diffusing pollen grains and Albert Einstein's statistical physics model that describes their motion. We analyze Jean Baptiste Perrin's data from Les Atomes using a probabilistic model to infer the uncertain diffusivities of the colloids. We show how the Bayesian paradigm allows us to assign and update our credences, before and after observing this data and quantify the information gained by the observation. In Chapter 2, we build on these concepts to provide insight on the phenomenon of enhanced enzyme diffusion, whereby enzymes are purported to diffuse faster in the presence of their substrate. We develop a hierarchical model of enzyme diffusion that describes the stochastic dynamics of individual enzymes drawn from a dispersed population. Using this model, we analyze single molecule imaging data of urease enzymes to infer their uncertain diffusivities for different substrate concentrations. Our analysis emphasizes the important role of model criticism for establishing self-consistency between experimental observations and model predictions; moreover, we caution against drawing strong conclusions when such consistency cannot be established. In Chapter 3, we automate, and optimize the data acquisition process, tuning a resonant acoustic cell using minimal experimental resources. By iterating a cycle of observation, inference, and design, we select the frequency the applied signal and the framerate of the data acquisition, garnering the same amount of information as a grid search approach with a fraction of the data. Finally, in Chapter 4, we discuss the role of Bayesian inference and design to optimize functional goals and discuss selected examples on where black-box techniques may prove useful. We review the current state of the art for magnetically actuated colloids and pose the search for autonomous magnetic behaviors as a design problem, offering insight as we seek to augment and accelerate the capabilities of micron scale magnetically actuated colloids using modern computational techniques.

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The general problem considered in this thesis is to determine an optimum strategy for deciding how to allocate the observations in each stage of a multi-stage experimental procedure between two binomial populations (e.g., the numbers of successes for two drugs) on the basis of the results of previous stages. After all of the stages of the experiment have been performed, one must make the terminal decision of which of the two populations has the higher probability of success. The optimum strategy is to be optimum relative to a given loss function; and a prior distribution, or weighting function, for the probabilities of success for the two populations is assumed. Two general classes of loss functions are considered, and it is assumed that the total number of observations in each stage is fixed prior to the experiment. In order to find the optimum strategy a method of analysis called extensive-form analysis is used. This is essentially a method for enumerating all the possible outcomes and corresponding strategies and choosing the optimum strategy for a given outcome. However, it is found that this method of analysis is much too long for all but small examples even when a digital computer is used. Because of this difficulty two alternative procedures, which are approximations to extensive-form analysis, are proposed. In the stage-by-stage procedure one assumes that at each stage he is at the last stage of his multi-stage procedure and allocates his observations to each of the two populations accordingly. It is shown that this is equivalent to assuming at each stage one has a one stage procedure. In the approximate procedure one (approximately) minimizes the posterior variance of the difference of the probabilities of success for the two populations at each stage. The computations for this procedure are quite simple to perform. The stage-by-stage procedure for the case that the two populations are normal with known variance rather than binomial is considered. It is then shown that the approximate procedure can be derived as an approximation to the stage-by- stage procedure when normal approximations to binomial distributions are used. The three procedures are compared with each other and with equal division of the observations in several examples by the computation of the probability of making the correct terminal decision for various values of the population parameters (the probabilities of success}. It is assumed in these computations that the prior distributions of the population parameters are rectangular distributions and that the loss functions are symmetric} i.e., the losses are as great for one wrong terminal decision as they are for the other. These computations show that, for the examples studied, there is relatively little loss in using the stage-by-stage procedure rather than extensive-form analysis and relatively little gain in using the approximate procedure instead of equal division of the observations. However, there is a relatively large loss in using the approximate procedure rather than the stage-by-stage procedure when the population parameters are close to 0 or 1. At first it is assumed there are a fixed number of stages in the experiment, but later in the thesis this restriction is weakened to the restriction that only the maximum number of stages possible in the experiment is fixed and the experiment can be stopped at any stage before the last possible stage is reached. Stopping rules for the stage-by- stage and the approximate procedures are then derived. / Ph. D.

In this thesis we consider the following. We choose the random variable θ, which has some fixed but unknown distribution with a finite second moment. We observe the value x, of a preliminary random variable X, which has an unknown distribution which is conditional on θ. Using x and our past experience we are asked to estimate the value of θ with a squared error loss function. After we have made our decision we are given the value, y, of a detailed random variable Y, which has an unknown distribution conditional on θ. The random variable X and Y are assumed independent given a particular θ. Our past experience is made up of the values of preliminary and detailed random variables from previous decision problems which are independent of but similar to the present one. With the risk defined in the usual way the Bayes decision function is the expected value of θ given that X = x. Since the distributions are unknown, the use of the two sample nonparametric empirical Bayes decision function is proposed. With the regret defined in the usual way it can be shown that the two sample nonparametric empirical Bayes decision function is asymptotically optimal, i.e. for a large number of past decision problems, the regret in using the two nonparametric empirical Bayes decision function tends to zero, and it is the main purpose of this thesis to verify this property by using a hypothetical numerical example. / Master of Science

The purpose of this study was to compare the effectiveness of the classical estimator with that of a supplementary sample non-parametric empirical Bayes estimator in detecting an out-of-control situation arising in statistical quality control work. The investigation was accomplished through Monte Carlo simulation on the IBM-7040/1401 system at the Virginia Polytechnic Institute Computing Center, Blacksburg, Virginia. In most cases considered in this study, the sole criterion for accepting or rejecting the hypothesis that the industrial process is in control was the location of the estimate on the control chart for fraction defectives. If an estimate fell outside the 30 control limits, that particular batch was said to have been produced by an out-of-control system. In other cases the concept of "runs" was included as an additional criterion for acceptance or rejection. Also considered were various parameters, such as the mean in-control fraction defectives, the mean out-of-control fraction defectives, the~first sample size, the standard deviation of the supplementary sample estimates, and the number of past experiences used in computing the empirical Bayes estimator. The Monte Carlo studies showed that, for almost any set of parameter values, the empirical Bayes estimator is much more effective in detecting an out-of-control situation than is the classical estimator. The most notable advantage gained by using the empirical Bayes estimator is that long-range lack of detection is virtually impossible. / M.S.

The purpose of this study is to consider two different types of estimators for reliability using the extreme value distribution as the life-testing model. First the unbiased minimum variance estimator is derived. Then the Bayes' estimators for the uniform, exponential, and inverted gamma prior distributions are obtained, and these results are extended to a whole class of exponential failure models. Each of the Bayes' estimators is compared with the unbiased minimum variance estimator in a Monte Carlo simulation where it is shown that the Bayes' estimator has smaller squared error loss in each case. The problem of obtaining estimators with respect to an exponential type loss function is also considered. The difficulties in such an approach are demonstrated. / Master of Science

This thesis considers two distinct aspects of the empirical Bayes decision problem. The first aspect considered is the problem or point estimation and hypothesis testing. The second aspect considered is that of estimating the prior distribution and then the estimation of posterior distribution and confidence intervals. In the first aspect considered we assume that there exists an unobservable parameter space 𝔏={λ} on which is defined a prior distribution G(λ). For any action a from a class A there is a loss, L(a,λ) ≥ 0, which we incur when we take action a and the true parameter is λ. There exists an observable random vector X̰=(X₁,...X_k), k ≥ 1, which admits of a sufficient statistic T=T(X̰) for λ. The conditional density function (c.d.f.) of T is f(t(λ). We assume that there exists a decision function δ_ɢ(t) from a class D (δεD) implies that δ(t)εA for all t) such that the expected loss, R(δ,G) = ∫∫L(δ(t),λ) f(t|λ)dtdG(λ), is minimized. This minimizing decision function is called a Bayes decision function and the associated minimum expected loss is called the Bayes risk R(G). We assume that there exists a sequence or independent identically distributed random vectors <(X₁,...,X_k,λ)_n> (or <(T,λ)_n >) with each element distributed independently of and identically to (X₁,...,X_𝗄,λ) (or (T,λ). The problem is to construct sequential decision functions, δ_n(t;t₁,t₂,...,t_n)=δ_n(t), which are asymptotically optimal (a.o.), that is which satisfy lim_n→∞ R(δ_n(T),G) = R(G). We extend a theorem or Robbins (Ann. Math. Statist. 35,1-20) to provide a simple method for the construction or a.o. point estimators of λ with a squared-error loss function when f(t|λ) is continuous. We extend the results or Samuel (Ann. Math. Statist., 34, 1370-1385) to provide a.o. tests of certain parametric hypotheses with loss functions which are polynomials in λ. The c.d.f.'s which are considered are all continuous and include some or those of the exponential class and some whose range depends upon the parameter. This latter result is applied to the problem or the one-sided truncation of density functions. The usefulness of all or these results is predicated upon the tact that the forms or the Bayes decision functions can be determined in such a way that the construction or the analogous a.o. empirical Bayes decision functions is simple. Two manipulative techniques, which provide the desired forms of the Bayes decision function, are introduced. These techniques are applied to several examples, and a.o. decision functions are defined. To estimate the prior distribution we assume that there exists a sequence of independent identically distributed random vectors <(T,λ)_n>) each distributed according to the joint density function J(t,λ)=G(λ)F(t|λ). The sequence <λ_n> of <(T,λ)_n> is unobservable. G(λ) belongs to a subclass g of a class G_p(g) and F(t|λ) belongs to a class F. G_p(g) is determined by the conditions: (a) G(λ) is absolutely continuous with with respect to Lebesgum measure; (b) its density function, g(λ), is determined completely by a continuous function of its first p moments, p ≥ 2; (c) the first p moments are finite; (d) the subclass g contains those density functions which are determined by a particular known continuous function. The class F is determined by the condition that there exist known functions h_𝗸(.), k=1,...,p, such that E[h_𝗸(T)|λ] = λᵏ. Under these conditions we construct an estimate, Gn(λ), of G(λ) such that lim_n→∞ E[(G_n(λ) - G(λ))²] = 0, a.e.λ. Estimates of the posterior distribution and of confidence intervals are constructed using G_n(λ). / Ph. D.

In a routinely occurring estimation problem, the experimenter can often consider the interested parameters themselves as random variables with unknown prior distribution. Without knowledge of the exact prior distribution the Bayes estimator cannot be obtained. However, as long as independent repetitions of the experiment occur, the empirical Bayes approach can then be applied. A general strategy underlying the empirical Bayes estimator consists of finding the Bayes estimator in a form which can be estimated sequentially by using the past data. Such use of the data circumvents knowledge of the prior distribution. Three different types of sampling distributions of cell counts of 2x2 contingency tables were considered. In the Independent Poisson Case, an empirical Bayes estimator for the cross-product ratio is presented. If the squared error loss function is used, this empirical Bayes estimator, ᾶ, has asymptotic risk only ε > 0 larger than the true Bayes risk. For the Product Binomial and Multinomial situations, several empirical Bayes estimators for α are proposed. Basically, these 'empirical' Bayes estimators by-pass the prior distribution by estimating the marginal probabilities P(X₁₁,X₂₁,X₁₂,X₂₂) and P(X₁₁+1,X₂₁-1,X₁₂-1,X₂₂+1), where (X₁₁,X₂₁,X₁₂,X₂₂) is the set of current cell counts. Furthermore, because of the assumption of varying sample size(s), they will have asymptotic risk only ε > 0 away from the true Bayes risk if both the number of past experiences and the sample size(s) are sufficiently large. Results of Monte Carlo simulation of empirical Bayes estimators are presented for the carefully selected prior distributions. Mean squared errors for these estimators and classical estimators were compared. The improvement of empirical Bayes over classical estimators was found to be dependent upon the prior means, the prior variances, the prior distribution of the parameters considered as random variables, and sample size(s). These conclusions are summarized, and tables are provided. The empirical Bayes estimators of α start to show significant improvement over classical estimators for as few as only ten past experiences. In many instances, the improvement is something on the order of 15% with only ten past experiences and sample size(s) larger than twenty. However, for the cases where the prior variances are very large, the empirical Bayes estimator indicates neither better nor worse over the classical. Greater improvement is shown for more past experiences until around thirty when the improvement appears stabilized. Finally, the other existing estimators for a which also take into account past experiences are discussed and compared to the corresponding empirical Bayes estimator. They were proposed respectively by Birch, Goodman, Mantel and Haenszel, Woolf, etc. The simulation study for comparisons indicate that empirical Bayes estimators outmatch them even with small prior variance. A test for deciding when empirical Bayes estimators of α should be used is also suggested and discussed. / Ph. D.